PULSE CONTROL OF SLIP IN INDUCTION MOTORS WITH ADDITIONAL RESISTANCES IN THE ROTOR CIRCUIT

Yu. P. Kostyukov, O. G. Makarchenko

(Presented by the Scientific and Technical Seminar of the Research Institute of AEM, TPI)

In recent years, in electric drives, the pulse method of controlling the slip of induction motors with a wound rotor at an increased switching frequency has found application. Pulse systems with additional resistances in the rotor circuit, unlike the rheostat method, make it possible to regulate the slip of an induction motor smoothly and without contact.

Figures 1 and 2 show the simplest schematic diagrams of pulse-controlled resistances in the rotor circuit of an induction motor \(AD\). The pulse-control circuits considered consist of an uncontrolled rectifier \(B\), assembled according to a three-phase bridge circuit; additional resistances connected into the rotor circuit of direct or alternating current; and a thyristor (transistor) switch \(K\).

Fig. 1.

Fig. 2.

The essence of the pulse method consists in a periodic change in the electrical parameters of the secondary circuit of the induction motor. When the switch \(K\) is closed, energy is stored in the motor inductances, and during the pause, under the action of the emf of self-induction and the emf

of the motor, current passes through the added resistances.

Determination of current pulsation. In pulse control of the slip of induction motors, current pulsations arise in the rotor circuit; their magnitude depends on the inductance of the intermediate circuits, the commutation frequency, the pulse duration, the control range, and the power-circuit configuration.

In systems with pulse-controlled resistance in the dc circuit, under generally accepted assumptions, the induction motor with rectifier is replaced by an equivalent generator. With a variable commutation frequency, the current pulsation coefficient is determined as follows:

\[ \frac{I_2}{I_1} = \frac{ 1-e^{-\frac{\alpha \beta \gamma}{1-\gamma}} +\beta e^{-\frac{\alpha \xi \gamma}{1-\gamma}}\left(1-e^{-\alpha}\right) }{ \left(1-e^{-\frac{\alpha \xi \gamma}{1-\gamma}}\right)e^{-\alpha} +\beta\left(1-e^{-\alpha}\right) }, \tag{1} \]

where

\[ I_2,\ I_1 \]

— respectively, the maximum and minimum values of the current of the established periodic process;

\[ \gamma=\frac{T-t_{\mathrm{u}}}{T} \]

— the relative duration of turn-on of the thyristor (transistor) switch \(K\);

\[ T \]

— commutation period;

\[ t_{\mathrm{u}} \]

— pulse duration corresponding to the time during which switch \(K\) is in the off state;

\[ \alpha=\frac{t_{\mathrm{u}}}{T_2},\quad \beta=\frac{T_2}{T_1} \]

— system parameters;

\[ T_1=\frac{1.75L+L_{\mathrm{dr}}}{R_{\mathrm{e}}S+1.75R'},\quad T_2=\frac{1.75L+L_{\mathrm{dr}}}{R_{\mathrm{e}}S+1.75R'+R_{\mathrm{u}}} \]

— time constants of the circuit during the first and second commutation intervals;

\[ L \]

— leakage inductance of a phase of the induction motor;

\[ R_{\mathrm{e}}=\frac{3}{\pi}\left(X'_1+X_2\right) \]

— equivalent ohmic resistance accounting for the voltage drop due to valve overlap at \(S=1\);

\[ S \]

— slip of the induction motor;

\[ R'=\left(R'_1\cdot S+R_2\right) \]

— active resistance of a phase of the induction motor;

\[ R'_1,\ R_2,\ X'_1,\ X_2 \]

— respectively, the active and inductive resistances of a phase of the induction motor, referred to the rotor circuit, at \(S=1\), determined from the refined Γ-shaped equivalent circuit with the stator and rotor windings connected in star;

\[ L_{\mathrm{dr}} \]

— inductance of the smoothing reactor.

Expanding expression (1) in a series and retaining the first two terms, which is quite permissible for small \(\gamma\), we obtain a simplified equation for the current pulsation coefficient:

\[ \frac{I_2}{I_1}=\frac{1-\alpha\beta\gamma}{1-\alpha\gamma}. \tag{2} \]

In pulse-control systems with additional resistances in an AC circuit, the range of pulsations varies and reaches a maximum at the amplitude value of the first harmonic of the phase current. In the latter case the current pulsation coefficient is determined by the expression:

\[ \frac{I_2}{I_1}= \frac{ 1-e^{-\frac{\alpha\gamma\beta}{1-\gamma}} +\beta\frac{1+\operatorname{tg}^2\varphi_1}{1+\beta^2\operatorname{tg}^2\varphi_1} \cdot e^{-\frac{\alpha\beta\gamma}{1-\gamma}}(1-e^{-\alpha}) }{ \left(1-e^{-\frac{\alpha\beta\gamma}{1-\gamma}}\right)e^{-\alpha} +\beta\frac{1+\operatorname{tg}^2\varphi_1}{1+\beta^2\operatorname{tg}^2\varphi_1}(1-e^{-\alpha}) }, \tag{3} \]

where \(I_2, I_1\) are, respectively, the maximum and minimum current values at the amplitude value of the first harmonic;

\[ \operatorname{tg}\varphi_1=2\pi f_c\cdot sT_1 \]

is a system parameter.

Here the time constants during the first and second commutation intervals are determined as follows:

\[ T_1=\frac{L}{R'},\qquad T_2=\frac{L}{R'+R_g}. \]

In systems with variable commutation frequency, the current pulsations have a maximum value at the greatest relative duration of switching-on and depend on the system parameters \(\alpha\), \(\beta\), and \(\operatorname{tg}\varphi_1\). As is evident from expressions (1) and (3), the current pulsations depend on the slip of the induction motor; therefore, in each specific case they are determined from the known mechanical characteristics and parameters of the electric-drive system.

As an example, Fig. 3 gives the dependences of the current pulsation coefficient on \(\alpha\) at \(\gamma=0.9\) and different \(\beta\), constructed from expression (1). From these curves one can easily determine the current pulsation coefficient, knowing the parameters of the pulse system.

Fig. 3

For given permissible current pulsations, depending on the control range (parameter \(\beta\)), from the expressions given we find

the required value of the parameter $a$. From the known $a$ and $T_2$ we determine the required pulse duration $t_u$.

Mechanical characteristics. Regulation of the slip of an induction motor is carried out by changing the relative duration of switching-on. The mechanical characteristics of pulse systems with additional resistances in the rotor circuit have the same form as in the rheostat method.

In pulse regulation with additional resistances in the rotor circuit of direct current, the artificial mechanical characteristics can be calculated by the usual methods used in the theory of cascade circuits, applying the concept of an equivalent additional resistance $R_u(1-\gamma)$ introduced into the rotor circuit [1].

In pulse-control circuits with additional resistances in the alternating-current circuit, uncontrolled valves operate in the pulse mode, there is no pause in the phase-current curve, and the calculation of the mechanical characteristics is simplified. In this case, for small current pulsations the maximum motor torque may be regarded as unchanged and equal to the torque of a single induction motor, and the mechanical characteristics may be calculated by the Kloss formula. Under the given conditions, the critical slip is equal to the sum of the critical slip of a single induction motor on this artificial characteristic and the initial slip. The critical slip on the artificial characteristic is determined taking into account the equivalent resistance introduced into the rotor circuit, which is easily found from the energy balance and is equal to $R_g(1-\gamma)$. For $\gamma = 1$, the magnitude of the initial slip corresponds to the speed of the fictitious ideal no-load operation and is determined by the voltage drop across the valves. Then, for the critical slip of an induction motor, the following expression is valid:

\[ s_k = \frac{R'_2 + R'_g(1-\gamma)} {\sqrt{R_1^2 + X_k^2}} + \frac{k_e \gamma \Delta U}{2.34 U_1}, \tag{4} \]

where

\[ \begin{aligned} U_1 &\text{— rms value of the phase voltage of the supply network;}\\ \Delta U &\text{— direct voltage drop across three valves;}\\ k_e &\text{— transformation coefficient from stator to rotor;}\\ X_k = X_1 + X'_2 &\text{— inductive resistance of the phase of the induction motor.} \end{aligned} \]

In expression (4), all parameters are referred to the stator of the induction motor.

Power factor and components of the motor apparent power. In the general case, the power factor of periodic nonsinusoidal currents is determined from the equality:

\[ \psi = \frac{P_a}{\sqrt{P_a^2 + P_p^2 + P_\nu^2}}, \tag{5} \]

where $P_a$, $P_p$, $P_\nu$ are, respectively, the active, reactive, and distortion power.

Let us consider the components of apparent power and the power factor in pulse systems when the additional resistances are connected in the direct-current circuit.

With sinusoidal primary voltage, only the active component of the first harmonic of the current participates in the production of active power.

\[ P_a = 3U_1 I_c \cos \varphi . \tag{6} \]

The reactive power consumed by the motor is due to the reactive component of the fundamental harmonic of the current and the reactive power

of the main field. Neglecting the active component of the no-load current of the induction motor, we obtain

\[ P_{\mathrm{p}} = 3U_1\left(I_{\mathrm{c}}\cdot \sin\varphi + I_0\right), \tag{7} \]

where

\(I_0\) is the value of the no-load current at slip \(s=0\).

In the expressions given, \(I_{\mathrm{c}}\) and \(\cos\varphi\) are determined by the formulas [1]:

\[ I_{\mathrm{c}}=\frac{4\sqrt{3}\,I_1\sin\frac{\lambda}{2}}{\sqrt{2}\,\pi\lambda\sigma k_{\mathrm{e}}}, \tag{8} \]

\[ \varphi=\frac{I_{\mathrm{c}}}{U_1}\sqrt{R_1^2+X_1^2}\, \sin\left(\operatorname{arctg}\frac{X_1}{R_1}-\frac{\lambda}{2}\right)+\frac{\lambda}{2}, \tag{9} \]

where

\(\lambda\) is the valve overlap angle, determined in the usual way;
\(\sigma\) is the leakage coefficient.

All higher harmonics contained in the primary current take part in producing the distortion power

\[ P_{\upsilon}=3U_1\sqrt{\sum_{l\ne 1}^{\infty}(i_l)^2}. \tag{10} \]

Assuming that the distortion coefficient of the secondary and primary currents is the same, we obtain:

\[ P_{\upsilon}=3U_1 I_{\mathrm{c}}\sqrt{\frac{1}{\upsilon^2}-1}. \tag{11} \]

The distortion coefficient \(\upsilon\) varies within small limits and is determined by the expression given in [2]. The power factor of the induction motor is found from the formula:

\[ \Psi= \frac{\cos\varphi}{ \sqrt{ \frac{1}{\upsilon^2} +\frac{I_0^2}{I_{\mathrm{c}}^2} +\frac{2I_0}{I_{\mathrm{c}}}\sin\varphi } }. \tag{12} \]

In practical calculations one may take \(\varphi \simeq \dfrac{\lambda}{2}\); then

\[ \Psi= \frac{\cos\dfrac{\lambda}{2}}{ \sqrt{ \frac{1}{\upsilon^2} +\frac{I_0^2}{I_{\mathrm{c}}^2} +\frac{2I_0}{I_{\mathrm{c}}}\sin\dfrac{\lambda}{2} } }. \tag{13} \]

With pulse control with additional resistances in the direct-current circuit, the power factor of the induction motor is always lower than with conventional connection, owing to valve overlap and distortion of the motor current curves.

In pulse systems with additional resistances in the alternating-current circuit, at a high commutation frequency the distortion power may be neglected. Then, on the basis of the refined \(\Gamma\)-shaped equivalent circuit of an induction machine, the active and reactive powers are determined as follows:

\[ P_{\mathrm{a}}=\frac{3U_1^2R}{R^2+X_{\mathrm{k}}^2}, \tag{14} \]

\[ P_p = \frac{3U_1^2 X_k}{R^2 + X_k^2} + \frac{3U_1^2}{X_o}, \tag{15} \]

where \(X_o\) is the inductive reactance of the magnetizing circuit;

\[ R = R_1 + \frac{R'_2 + R'_g(1-\gamma)}{s}. \]

The power factor of the induction motor will in practice be the same as with rheostatic regulation, and is determined from the expression

\[ \Psi = \frac{1}{ \sqrt{ 1+ \left[ \frac{ X_k\left(1+\dfrac{X_k}{X_o}\right) }{R} + \frac{R}{X_o} \right]^2 } }. \tag{16} \]

On the whole, in terms of energy indices the pulse systems under consideration are only slightly inferior to the rheostatic method. Under identical conditions in both pulse-control systems, the overload capacity of the motor, the distortion coefficients, and the power in circuits with additional resistances in the a.c. circuit are always greater.

The advantages of the pulse method of regulation with additional resistances include: the simplicity of the pulse system, high response speed and the possibility of stepless starting, smooth slip regulation, and a comparatively high efficiency under fan load and within a limited regulation range.

REFERENCES

1. Yu. P. Kostyukov. Development and investigation of pulse systems for regulating the rotational speed of induction motors with a wound rotor. Author’s abstract of dissertation. Tomsk, 1967.

2. Yu. P. Kostyukov. Certain questions in the theory of induction motors with a wound rotor under pulse speed regulation. Izvestiya TPI, 1967, vol. 161.

3. M. P. Kostenko, L. M. Piotrovsky. Electrical Machines, part 2, GEI, 1958.

4. A. V. Posse. Justification of replacing a rectifier by an equivalent generator for calculating transient processes. Izvestiya AN SSSR, “Energetika i transport,” 1965, No. 4.